NUMERICAL DESCRIPTIVE STATISTICS Measures of Central Tendency Measures of Central Tendency

MEASURES OF CENTRAL TENDENCY The following are typical measures of central tendency for a population –Mean -- the average –Median -- the middle observation after the data has been ordered –Mode -- the observation that occurs most often

One measure of central tendency is the mean The population mean is the average (or weighted average) of all observations of the population Population mean for a population of size N:  =  x i /N Population Mean (  )

Example N = 2000 students took an introductory statistics last year at CSUF. Using the 4- point scale (A=4, B=3, etc.) the following were the grades 4,2,1,3,3,3,2,… 2. The mean grade of all statistics students was:  = ( …+2)/2000 = 2.39

Calculating  Using Frequency Data Would not add up the 2000 numbers this way if we knew how many A’s, B’s, C’s, D’s, and F’s Example A(4) = 304, B(3) = 530, C(2) = 852, D (1) = 270 F(0) = 44  = (304(4) +530(3) +852(2) +270(1) +44(0))/2000 = 2.39 Note the relative frequencies are found by dividing by N (which is 2000): A(4) =.152, B(3) =.265, C(2) =.426, D (1) =.135 F(0) =.022 We can also calculate  as a weighted average (weighted by relative frequencies) :  =.152(4) +.265(3) +.426(2) +.135(1) +.022(0) = 2.39

Sample Data The only way to know  for sure is to have access to all the data in the population We rarely know all the members of the population Thus we try to estimate a parameter such as  by taking a sample of n members from the N members of the population POINT ESTIMATEResult: POINT ESTIMATE of the parameter

EXAMPLE Suppose 10 students were surveyed. Their grades were: 4,2,3,3,2,2,1,4,3,2

Sample Mean The sample mean is the point estimate for the population mean. DATA: 4,2,3,3,2,2,1,4,3,2

Frequency Calculation for the Sample Mean Grade Frequency A B C D

Sample Median (Even Number of Data Points) The the average of the two middle observations First put in either ascending or descending order DATA: 4,2,3,3,2,2,1,4,3,2 There is an even number of data points (10) ASCENDING ORDER: 1,2,2,2,2,3,3,3,4,4 2,3 Median = Average of the middle two (2+3)/2 = 2.5

Sample Median (Odd Number of Data Points) Suppose there were an odd number of observations -- suppose an 11 person was surveyed and she got an “A” (4) There is an odd number of data points (11) ASCENDING ORDER: 1,2,2,2,2,3,3,3,4,4,4 3 Median is middle observation Median = 3

Observation that occurs most often in the sample data Create a frequency distribution Grade Frequency A B C D Sample Mode 2 Mode = 2 4 Highest Frequency

Central Tendency in Excel Suppose data are in cells A2 to A11 Mean --=AVERAGE(A2:A11) Median -- =MEDIAN(A2:A11) Mode --=MODE(A2:A11) Can also use Descriptive Statistics Option from Data Analysis in the Tools Menu

=AVERAGE(A2:A11)=MEDIAN(A2:A11) =MODE(A2:A11)

Where data values are stored Enter Name of Output Worksheet Check both: Summary Statistics Confidence Level Check Labels

Sample Mean Sample Median Sample Mode Drag to make Column A wider

Review Difference Between Population and Sample Data Calculation of the Mean: –Definition –Relative Frequency Approach Median Mode Excel –Functions –Descriptive Statistics